On Numerical Approaches for Solving an Inverse Cauchy Stokes Problem

In this paper, we are interested in the study of an inverse Cauchy problem governed by Stokes equation. It consists in determining the fluid velocity and the flux over a part of the boundary, by introducing given measurements on the remaining part. As it’s known, it is one of highly ill-posed problems in the Hadamard’s sense (Phys Today 6:18, 1953), it is then an interesting challenge to carry out a numerical procedure for approximating their solutions, in particular, in the presence of noisy data. To solve this problem, we propose here a regularizing approach based on a Tikhonov regularization method. We show the existence of the regularization optimization problem and prove the convergence of subsequence of optimal solutions of Tikhonov regularization formulations to the solution of the Cauchy problem, when the noise level goes to zero. Then, we suggest the numerical approximation of this problem using the finite elements method of P1Bubble/P1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{1Bubble}/P_1$$\end{document} type’s, we show the existence of the discrete optimal regularized solution without noise and prove the convergence of subsequence of discrete optimal solutions to the solution of the continuous optimization problem. Finally, we provide some numerical results showing the accuracy and the efficiency of the proposed approach.